I am trying to create a motor driver circuit using a darlington driver chip with 4x NPN Darlington switches. I believe I know how to accomplish the circuit with 2x NPN and 2x PNP within a chip but is it also possible to do so with 4x NPN switches?
Possible yes (with limited joy).
Assuming your motor voltage is 12V, the high side NPN's would need a basis voltage of around 13.5V to conduct properly (which your logic can't deliver).
Besides, Darlingtons have a high voltage drop, so you may end up with around 3V less (than your supply) for the motor.
The chip I currently own is a ULN2064B.
Always post a link to a datasheet when possible - it gives you a better chance of getting useful answers.